KöMaL 2017 B4879

$a)$ Is it true that for any irrational number $a$ there exists an irrational number $x$ such that $a+x$ is rational and $ax$ is irrational? $b)$ Is it true that for any irrational number $a$ there exists an irrational number $y$ such that $a+y$ is irrational and $ay$ is rational?